Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle BOC = 4x + 18$, and $ m \angle AOB = 6x - 18$, find $m\angle BOC$. $O$ $A$ $C$ $B$
Explanation: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {6x - 18} + {4x + 18} = {90}$ Combine like terms: $ 10x + 0 = 90$ Add $0$ to both sides: $ 10x = 90$ Divide both sides by $10$ to find $x$ $ x = 9$ Substitute $9$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 4({9}) + 18$ Simplify: $ {m\angle BOC = 36 + 18}$ So ${m\angle BOC = 54}$.